Question

Solve differiantial equation (y+x) dy/dX=y-x

Answer 1

$(y+x) \frac{dy}{dx} = y-x \\ \\ Let, \left \{ {{y=vx} \atop { \frac{dy}{dx} =(v+x) \frac{dv}{dx}}} \right. \\ \\ (v+x) \frac{dv}{dx}} = \frac{vx-x}{vx+x} \\ (v+x) \frac{dv}{dx}} = \frac{v-1}{v+1} \\ x \frac{dv}{dx}} = \frac{v-1}{v+1} - v \\ x \frac{dv}{dx}} = \frac{v-1-v^2 -v}{v+1} \\ x \frac{dv}{dx}} = \frac{-(v^2+1)}{v+1}\\ \int\limits\frac{(v+1)}{v^2+1}dv=\int\limits- \frac{dx}{x}$

Now integrate both sides and you will get the answer.